Van der Waals interactions in the lattice of metallic chains

2014 The cohesive energy of a lattice of metallic chains is calculated as a perturbation expansion of the Coulomb interaction, which is assumed small relative to the intrachain overlap integrals. The expansion contains the Van der Waals like interactions associated with the metallic polarizabilities of the chains. For sufficiently large forward interchain coupling (backward contribution neglected), the Van der Waals energy gained on going from the HMTTF-TCNQ to TTF-TCNQ lattice competes with the corresponding loss of the Madelung energy of the charges homogeneously distributed along the chains. J. Physique LETTRES 44 (1983) L327 L332 ler m 1983 Classification Physics Abstracts 71. 45N 61. 50L Organic metals belonging to the TTF-TCNQ family which have been extensively studied in recent years crystallize in a segregated stack structure [1]. Like molecules TTF or HMTTF (F) and TCNQ (Q) form conducting chains along the b-axis. The organization of the chains in the a-c plane differs however according to the material. E.g. figure la shows schematically the a-c plane in TTF-TCNQ and figure lb the same plane in HMTTF-TCNQ. Due to the transfer [2, 3] of a fractional number p of electrons per donor or acceptor molecule ( p ~ 0.6-0.7) the chains are charged. This charge is evenly distributed along the chain in the metallic phase, but at low enough temperatures ( 10-10 2K ) the 2 kF, 4 k F modulations ( CDWs ) may appear [2,3] k F == ~ p The condensation of these superstructures has little or no effect on the charge transfer (i.e. on kF) itself [2, 3] and on the lattice parameters [4]. In most previous attempts to understand the occurrence of the segregated stack structures attention was focused on the balance between the ionization affmity energy E~ and the Madelung energy EM [5, 6]. It turned out [5, 6] that with an homogeneous charge distribution Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01983004409032700 L-328 JOURNAL DE PHYSIQUE LETTRES Fig. 1. Schematic packing of chains in the TTF-TCNQ (a) and HMTTF-TCNQ (b) structure. along the chain the energy gain EM was insufficient to offset the loss EIA. This energy imbalance may be improved by assuming [7, 8] that the transfered charges are not uniformly distributed, but form Wigner lattices along the chains. However, it should be realized that these lattices represent extreme cases of the 2 kF, 4 kF CDWs. If their formation were responsible for the stability of the charge transfer, their disappearance at low temperatures should be accompanied by significant variations of the charge transfer (i.e. of 2 kF) and of the lattice parameters. This is at variance with the experimental results mentioned above. Besides, arguments based solely on the Madelung and the ionization energies should lead to full charge alternation in all directions (i.e. to the Q-F alternation). A different mechanism is therefore required to understand the stability of the segregated stack structures. In this respect Friedel has pointed out [9] that the dispersive forces (Van der Waals interactions, interactions of permanent dipoles etc.) favour the close packing of like molecules. His argument can be reproduced in a few words : consider [10] the two molecular distributions of figures la, b. Going from figure 16 to figure la involves replacing the two Q-F bonds by a Q-Q and an F-F bond. Assume for example that those are the Van der Waals (VdW) bonds. Keeping all other parameters constant and using the London approximation for the Van der Waals energy, the corresponding energy change is as required. Here a’s denote the polarizabilities of the Q, F molecules. The argument (1), if carried to its extreme, leads to the full Q-F segregation. As the opposite extreme (full Q-F alternation) follows from the Madelung argument, it is tempting to attribute the observed lattices to the fine balance between the dispersive and the Madelung forces : in contrast to the Madelung forces the VdW forces prefer the TTF-TCNQ lattice of figure la to the HMTTF-TCNQ lattice of figure lb. Unfortunately, the homogeneous charge distribution along the chain is not a natural consequence of the model based exclusively on these two types of forces. First, the molecular polarizabilities seem to be too small to make the cohesive energy of this structure positive [7]. Second, the model tends to prefer the formation of the strong Wigner lattices along the chain to the homogeneous charge distribution. On the other hand, the homogeneous charge distribution is a natural ingredient of the model of metallic chains. Although widely accepted in the description of the low temperature properties [11-13], this model was not previously considered successful [5, 6] in the evaluations of the L-329 METALLIC VAN DER WAALS INTERACTIONS cohesive energy. The reason is that the band energy EB tends to be small with respect to EIA. However, we wish to point out here that there are other energy contributions arising from the metallic chains, which favour the observed lattices. In particular, we will show that the metallic chains act as polarizable units and the notion of VdW interactions between chains can be retained. The organization of the a-c plane in thus still covered by Friedel’s equation (1), upon identifying the a’s with the polarizabilities of the metallic chains. The condition for the validity of equation (1) in such an interpretation is that the interchain interactions are weak. Unfortunately, this requirement is not very well fulfilled in the actual materials. The Coulomb forces are long ranged, i.e. of the same order of magnitude within and between the chains. Their order of magnitude can be estimated [13] from the intra-band plasmon frequency c~o. As the latter turns out to be of the same order of magnitude as the band-width [14], we are encountering the intermediate coupling regime. Nevertheless, in order to conserve explicitly the attractive features associated with equation (1) we shall scale down all the interactions, rather than assume artificially that only the interchain couplings are weak. Solving the weak coupling problem we shall see that it is physically reasonable to extrapolate the results to moderate couplings. The method which bridges best between weak and strong coupling limit is the tight-binding (TB) method [15, 16]. From the point of view of Coulomb forces this method is related [16] to the Wigner-Seitz (W-S) approach [17]. The TB wave-functions satisfy the W-S boundary conditions and are calculated on each site with the appropriate ionic potential. Thereby they tend to anticipate the results of the RPA calculation [17]. In the TB/W-S scheme the cohesive energy is decomposed as EIA and EB have been already defined. EEc involves all the energies not related to the band formation and the charge transfer (includes e.g. « steric » effects). ESFE is the self-energy of the valence charge on the site. It brings us back from the W-S to the Hartree scheme, which is the usual starting point for the calculation of the Hartree-Fock energy EHF and the correlation energy E~op In ordinary metals the two last terms in equation (2) roughly cancel out ESFE, justifying the use of the « single electron » W-S term Ep [17]. The situation is somewhat different here. ESFE is the local (on-site, on-molecule) term, whereas EHF and EeOR contain the interchain terms, which we believe to be responsible for the choice between the two lattices in figure 1. A few low-order diagrams of the weak-coupling theory for EHF + E~oR are shown in figure 2. The term EH in figure 2a is the Hartree term, which is identical to the Madelung energy EM. Indeed, the loops in figure 2a denote the homogeneous charge distributions on the chains n and m which interact through the Coulomb interaction Y,~(q = 0). EF of figure 2b is the Fock term. This is an on-chain term, which cancels for the energy difference of the two lattices shown in figures la, b. The term which is of particular interest here is shown in figure 2c. For chains at distances Rnm we get from this figure We note immediately that the VdW energy between two chains [18] Evdw(r4 m) has the same general structure as the terms in equation (1). The diagram 2c (i.e. Eq. (3 )), which is usually interpreted as describing the short range (R ~ kTF1) forces between electrons of the jellium [19] characL-330 JOURNAL DE PHYSIQUE LETTRES